Table of Contents Author Guidelines Submit a Manuscript
Abstract and Applied Analysis
Volume 2013, Article ID 601325, 13 pages
http://dx.doi.org/10.1155/2013/601325
Research Article

Weakly Compact Uniform Attractor for the Nonautonomous Long-Short Wave Equations

School of Mathematics and Information, Ludong University, Yantai, 264025, China

Received 8 November 2012; Revised 25 January 2013; Accepted 27 January 2013

Academic Editor: Lucas Jódar

Copyright © 2013 Hongyong Cui et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. R. H. J. Grimshaw, “The modulation of an internal gravity-wave packet, and the resonance with the mean motion,” vol. 56, no. 3, pp. 241–266, 1976/77. View at Google Scholar · View at MathSciNet
  2. D. R. Nicholson and M. V. Goldman, “Damped nonlinear Schrödinger equation,” The Physics of Fluids, vol. 19, no. 10, pp. 1621–1625, 1976. View at Publisher · View at Google Scholar · View at MathSciNet
  3. D. J. Benney, “A general theory for interactions between short and long waves,” vol. 56, no. 1, pp. 81–94, 1976/77. View at Google Scholar · View at MathSciNet
  4. B. L. Guo, “The global solution for one class of the system of LS nonlinear wave interaction,” Journal of Mathematical Research and Exposition, vol. 7, no. 1, pp. 69–76, 1987. View at Google Scholar · View at MathSciNet
  5. B. Guo, “The periodic initial value problems and initial value problems for one class of generalized long-short type equations,” Journal of Engineering Mathematics, vol. 8, pp. 47–53, 1991. View at Google Scholar
  6. X. Du and B. Guo, “The global attractor for LS type equation in 1,” Acta Mathematicae Applicatae Sinica, vol. 28, pp. 723–734, 2005. View at Google Scholar
  7. R. Zhang, “Existence of global attractor for LS type equations,” Journal of Mathematical Research and Exposition, vol. 26, no. 4, pp. 708–714, 2006. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  8. Y. Li, “Long time behavior for the weakly damped driven long-wave-short-wave resonance equations,” Journal of Differential Equations, vol. 223, no. 2, pp. 261–289, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  9. B. Guo and L. Chen, “Orbital stability of solitary waves of the long wave-short wave resonance equations,” Mathematical Methods in the Applied Sciences, vol. 21, no. 10, pp. 883–894, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  10. B. Guo and B. Wang, “Attractors for the long-short wave equations,” Journal of Partial Differential Equations, vol. 11, no. 4, pp. 361–383, 1998. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  11. B. Guo and B. Wang, “The global solution and its long time behavior for a class of generalized LS type equations,” Progress in Natural Science, vol. 6, no. 5, pp. 533–546, 1996. View at Google Scholar · View at MathSciNet
  12. D. Bekiranov, T. Ogawa, and G. Ponce, “On the well-posedness of Benney's interaction equation of short and long waves,” Advances in Differential Equations, vol. 1, no. 6, pp. 919–937, 1996. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  13. D. Bekiranov, T. Ogawa, and G. Ponce, “Interaction equations for short and long dispersive waves,” Journal of Functional Analysis, vol. 158, no. 2, pp. 357–388, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  14. Z. Huo and B. Guo, “Local well-posedness of interaction equations for short and long dispersive waves,” Journal of Partial Differential Equations, vol. 17, no. 2, pp. 137–151, 2004. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  15. M. Tsutsumi and S. Hatano, “Well-posedness of the Cauchy problem for Benney's first equations of long wave short wave interactions,” Funkcialaj Ekvacioj, vol. 37, no. 2, pp. 289–316, 1994. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  16. M. Tsutsumi and S. Hatano, “Well-posedness of the Cauchy problem for the long wave-short wave resonance equations,” Nonlinear Analysis. Theory, Methods & Applications A: Theory and Methods, vol. 22, no. 2, pp. 155–171, 1994. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  17. R. Zhang and B. Guo, “Global solution and its long time behavior for the generalized long-short wave equations,” Journal of Partial Differential Equations, vol. 18, no. 3, pp. 206–218, 2005. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  18. V. V. Chepyzhov and M. I. Vishik, “Non-autonomous evolutionary equations with translation-compact symbols and their attractors,” Comptes Rendus de l'Académie des Sciences, vol. 321, no. 2, pp. 153–158, 1995. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  19. V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics, vol. 49 of American Mathematical Society Colloquium Publications, American Mathematical Society, Providence, RI, USA, 2002. View at MathSciNet
  20. V. V. Chepyzhov and M. I. Vishik, “Attractors of nonautonomous dynamical systems and their dimension,” Journal de Mathématiques Pures et Appliquées, vol. 73, no. 3, pp. 279–333, 1994. View at Google Scholar · View at MathSciNet
  21. J. Bergh and J. Laöfstraöm, Interpolation Spaces, Springer-Verlag, Berlin, Germany, 1976.
  22. J. Xin, B. Guo, Y. Han, and D. Huang, “The global solution of the (2+1)-dimensional long wave-short wave resonance interaction equation,” Journal of Mathematical Physics, vol. 49, no. 7, pp. 073504–07350413, 2008. View at Publisher · View at Google Scholar · View at MathSciNet